Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,18,2}

Atlas Canonical Name {4,6,18,2}*1728b

Overview

Group
SmallGroup(1728,30872)
Rank
5
Schläfli Type
{4,6,18,2}
Vertices, edges, …
4, 12, 54, 18, 2
Order of s0s1s2s3s4
36
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);;
s1 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)(188,212)(189,213);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 54)( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)(116,117)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)(125,134)(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)(164,168)(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)(207,214);;
s3 := (  1,118)(  2,120)(  3,119)(  4,124)(  5,126)(  6,125)(  7,121)(  8,123)(  9,122)( 10,109)( 11,111)( 12,110)( 13,115)( 14,117)( 15,116)( 16,112)( 17,114)( 18,113)( 19,129)( 20,128)( 21,127)( 22,135)( 23,134)( 24,133)( 25,132)( 26,131)( 27,130)( 28,145)( 29,147)( 30,146)( 31,151)( 32,153)( 33,152)( 34,148)( 35,150)( 36,149)( 37,136)( 38,138)( 39,137)( 40,142)( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,156)( 47,155)( 48,154)( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,172)( 56,174)( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)( 62,177)( 63,176)( 64,163)( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)( 70,166)( 71,168)( 72,167)( 73,183)( 74,182)( 75,181)( 76,189)( 77,188)( 78,187)( 79,186)( 80,185)( 81,184)( 82,199)( 83,201)( 84,200)( 85,205)( 86,207)( 87,206)( 88,202)( 89,204)( 90,203)( 91,190)( 92,192)( 93,191)( 94,196)( 95,198)( 96,197)( 97,193)( 98,195)( 99,194)(100,210)(101,209)(102,208)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211);;
s4 := (217,218);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(218)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);
s1 := Sym(218)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)(188,212)(189,213);
s2 := Sym(218)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 54)( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97,108)( 98,107)( 99,106)(109,112)(110,114)(111,113)(116,117)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,135)(125,134)(126,133)(136,139)(137,141)(138,140)(143,144)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,162)(152,161)(153,160)(163,166)(164,168)(165,167)(170,171)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,189)(179,188)(180,187)(190,193)(191,195)(192,194)(197,198)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,216)(206,215)(207,214);
s3 := Sym(218)!(  1,118)(  2,120)(  3,119)(  4,124)(  5,126)(  6,125)(  7,121)(  8,123)(  9,122)( 10,109)( 11,111)( 12,110)( 13,115)( 14,117)( 15,116)( 16,112)( 17,114)( 18,113)( 19,129)( 20,128)( 21,127)( 22,135)( 23,134)( 24,133)( 25,132)( 26,131)( 27,130)( 28,145)( 29,147)( 30,146)( 31,151)( 32,153)( 33,152)( 34,148)( 35,150)( 36,149)( 37,136)( 38,138)( 39,137)( 40,142)( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,156)( 47,155)( 48,154)( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,172)( 56,174)( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)( 62,177)( 63,176)( 64,163)( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)( 70,166)( 71,168)( 72,167)( 73,183)( 74,182)( 75,181)( 76,189)( 77,188)( 78,187)( 79,186)( 80,185)( 81,184)( 82,199)( 83,201)( 84,200)( 85,205)( 86,207)( 87,206)( 88,202)( 89,204)( 90,203)( 91,190)( 92,192)( 93,191)( 94,196)( 95,198)( 96,197)( 97,193)( 98,195)( 99,194)(100,210)(101,209)(102,208)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211);
s4 := Sym(218)!(217,218);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;